Optimal subsampling algorithms for composite quantile regression in massive data

Abstract

Massive datasets have gained increasing prominence across various fields, but their analysis is often impeded by computational limitations. In response, Wang and Ma (Optimal subsampling for quantile regression in big data. Biometrika. 2021;108:99–112) have proposed an optimal subsampling method for quantile regression in massive datasets. Composite quantile regression, as a robust and efficient alternative to ordinary least squares regression and quantile regression in linear models, presents further complexities due to its distinct loss function. This paper extends the optimal subsampling method to accommodate composite quantile regression problems. We begin by deriving two new optimal subsampling probabilities for composite quantile regression, considering both the L- and A-optimality criteria Subsequently, we develop an adaptive two-step method based on these probabilities. The resulting estimators exhibit desirable asymptotic properties. In addition, to estimate the variance-covariance matrix without explicitly estimating the densities of the responses, we propose a combining subsamples method. Numerical studies on simulated and real data are conducted to assess and showcase the practical performance of our proposed methods.